Question

If the line $y=x+k$ touches the circle $x^2+y^2=4,$ then find the value of $k$

Answer 1

The given circle is $x^2+y^2=4$ ..........$\left(1\right)$

Its centre is $C(0,0)$ and its radius is $2$

The given line is $x-y+k=0$ ..........$\left(2\right)$

The line $\left(2\right)$ will touch the circle $\left(1\right)$ if the perpendicular distance from $C(0,0)$ to the line $\left(2\right)=$radius of circle.

$\Rightarrow \frac{|1\times 0-0+k|}{\sqrt{1^2+1^2}}=2$

$\Rightarrow \frac{\left|k\right|}{\sqrt{2}}=2$

$\Rightarrow k=\pm 2\sqrt{2}$